Neste trabalho, estudamos modelos para crescimento de superfície. Mais especificamente, trabalhamos com um modelo que respeita a condicão RSOS e outro modelo que apresenta uma transicão de rugosidade da classe de universalidade da percolacão direcionada. Obtivemos resultados com aproximacão de campo médio e simulacão. / In this work we studied models for surface growth. More specifically, we worked with a model that presents the RSOS restriction and another one that displays a depinning transition in the direct percolation universality class. We obtained results using mean field approximation and simulation methods.

Neste trabalho, estudamos modelos para crescimento de superfície. Mais especificamente, trabalhamos com um modelo que respeita a condicão RSOS e outro modelo que apresenta uma transicão de rugosidade da classe de universalidade da percolacão direcionada. Obtivemos resultados com aproximacão de campo médio e simulacão. / In this work we studied models for surface growth. More specifically, we worked with a model that presents the RSOS restriction and another one that displays a depinning transition in the direct percolation universality class. We obtained results using mean field approximation and simulation methods.

Understanding eco-evolutionary dynamics in cancer tumors, species invasions, and the human microbiome is vital for numerous health and economic applications. However, spatial structure and population heterogeneity make this challenging. This dissertation tackles these challenges using a population dynamics approach, wherein systems evolve through individual growth and dispersal. The bulk of this dissertation studies expanding populations, such as growing microbial colonies, species range expansions, and cancer tumors. In this context, I first study the effect of a directional bias in dispersal: I develop a model for the stochastic growth of left-right or chirally asymmetric cells that quantitatively reproduces experimental patterns in microbial colonies. Using the model, I demonstrate that chiral dispersal provides an evolutionary advantage and affects spatial population structure in expanding populations. Second, I investigate the impact of environmental structure affecting both dispersal and growth on expanding populations. I show that cooperative population expansions in a periodic environment can be pinned to a particular location or locked to specific velocities determined by the environmental periodicity. Third, I study the problem of a phenotypically heterogeneous population, with each phenotype differing in growth and dispersal abilities. I determine the exact velocity of an expanding population where phenotypes move ballistically and explain the connection to the explosive growth transition in experimental microtubule asters. The final chapter of the dissertation examines the challenge of assembling microbial communities for performing functions such as biofuel production, nitrogen fixation, or health remediation. Due to the exponential number of possible species combinations, bioengineers resort to heuristic search strategies to find the optimal community. I identify biological properties and develop statistical measures to help bioengineers estimate their chance of success in assembling an optimal community. / 2023-02-06T00:00:00Z

The surface width, a global quantity that depends on time, is used to characterize the temporal evolution of growing surfaces. One of the most successful concepts for describing the property of the surface width is the famous Family-Vicsek scaling relation. We discuss an extended scaling relation that yields a complete description for various growth models. For two linear Langevin equations, namely the Edwards-Wilkinson equation and the Mullins-Herring equation, we furthermore study analytically the behavior of global quantities related to the surface width or to a quantity which is conjugated to the diffusion constant. The global quantities depend in a non-trivial way on two different times. We discuss the dynamical scaling forms of global correlation and response functions. For global functions related to the surface width, we show that the scaling behavior of the response can depend on how the system is perturbed. Different dynamic regimes, characterized by a power-law or by an exponential relaxation, are identified, and a dynamic phase diagram is constructed. We discuss global fluctuation-dissipation ratios and how to use them for the characterization of non-equilibrium growth processes. We also numerically study the same two-time quantities for the non-linear Kardar-Parisi-Zhang equation. For global functions related to the quantity which is conjugated to the diffusion constant of the linear Langevin equations, we show that the integrated response is proportional to the correlation in the linear response regime. In the aging regime, the autocorrelation and autoresponse exponents are identical and the aging exponent for the response is equal to the aging exponent for the correlation. We investigate the non-equilibrium fluctuation-dissipation theorem for non-equilibrium states based on this quantity. In the non-linear response regime a certain dissipation-fluctuation ratio approximates unity for small waiting times but approaches the ratio of perturbed and unperturbed diffusion constants for larger waiting times. / Ph. D.

Nesse trabalho, são investigadas as propriedades dinâmicas de modelos da mecânica estatística na criticalidade. Inicialmente, trabalhando com modelos de spin e utilizando os conceitos de persistência global e de dimensão anômala da magnetização inicial, mostramos que o modelo de Baxter-Wu não está na mesma classe de universalidade dos modelos de Potts com quatro estados e de Ising com interação de três spins em uma direção, todos bidimensionais. Na segunda parte da tese, estudamos o fenômeno de crescimento da superfície gerada pela deposição segundo as regras que definem os autômatos celulares probabilísticos propostos por Grassberger (modelos A e B). Esses dois autômatos não pertencem à classe de universalidade de Domany-Kinzel e apresentam novos expoentes críticos, cuja origem se deve à conservação de paridade. Determinamos o expoente de crescimento beta w, válido em tempos curtos, assim como os outros expoentes críticos associados ao crescimento de superfície (alfa e z). Nossas estimativas se comparam bem com os resultados obtidos a partir de razões de inteiros propostas por Jensen para os expoentes beta, ni paralelo e ni perpendicular. Finalmente, investigamos a transição de fase entre o estado helicoidal e o estado desordenado (random coil) da polialanina e do fragmento peptídico PTH(1-34), que corresponde aos resíduos 1 a 34 da região aminoterminal do hormônio das paratireóides. Nosso cálculo, que leva em conta as interações entre todos os átomos da molécula, está baseado em uma abordagem de tempos curtos. Os resultados dessa análise indicam que a transição helix-coil das polialaninas e do PTH(1-34) é de segunda ordem e apontam para uma classe de universalidade para a transição helix-coil em homopolímeros e proteínas (partindo de um estado helicoidal). / In this work we investigated dynamic properties of statistical mechanical models at criticality. At first, using the concepts of global persistence and anomalous dimension of initial magnetization, we showed that the Baxter-Wu model does not belong to the same universality class as 4-state Potts model and Ising with multispin interaction in one direction. In the sequence, we studied the roughening behavior generated by deposition governed by rules defined by probabilistic cellular automata proposed by Grassberger (A and B models). Those models are known do not belong to the Domany-Kinzel universality class. They are characterized by different exponents which are related to the parity conserving (PC). We estimated the growth exponent beta w, in short-time regimen, such as, other critical exponents associated to the surface growth (alpha and z). Our results are in good agreement with those expected for parity conserving universality class. At last we studied the phase transition between the completely helical state and the random coil of the polyalanine, such as, for the 34-residue human parathyroid fragment PTH(1-34). Our short-time simulations of the helix-coil transition are based on a detailed all-atom representation of proteins. The results indicate that helix-coil transition in polyalanine and PTH(1-34) is a second-order phase transition and suggest a universality class to the helix-coil transition in homopolymer and (helical) proteins.

Uno de los aspectos más fascinantes del mundo que nos rodea es la gran variedad de escalas a las que tienen lugar los diversos fenómenos. En muchos casos esta diversidad pone de manifiesto la estructura fractal de la Naturaleza y podemos hablar entonces de fenómenos complejos, en los que eventos de diferentes magnitudes no pueden analizarse de manera independiente. Dicha complejidad emerge como un fenómeno cooperativo a escalas microscópicas, que produce un complejo comportamiento macroscópico caracterizado por correlaciones de largo alcance e invarianza de escala. Aparecen así conceptos como leyes de escalado, universalidad y renormalización, pilares fundamentales dentro de la Física Estadística.El abanico de fenómenos complejos es muy amplio, y abarca sistemas de muy diversas disciplinas que van desde la Físicamás ortodoxa hasta la Biología, Sociología, Geología e, incluso, Economía. Esta Tesis se centra en fenómenos complejos extendidos en el espacio. En concreto hemos focalizado nuestra labor en tres grandes temas que constituyen importantes focos de interés dentro de la Mecánica Estadística: Crecimiento de Interfases, Sociofísica y Redes Complejas. / The ubiquity of complexity in Nature provides examples of a huge variety of systems to be analyzed by means of Statistical Mechanics and leads to the interconnection among various scientific disciplines. This Thesis focuses on three highlight topics of spatially extended complex systems: Interface Growth,Sociophysics, and Complex Networks. The document has been partitioned in three separated parts according to those topics.The first part deals with far-from-equilibrium growing interfaces. This subject represents one of the main fields in which fractal geometry has been widely applied, and is nowadays of great interest in Condensed Matter Physics. The Chapter 2 provides a brief and basic introduction to interface growth. We introduce some fractal and scaling concepts, as well as the main universality classes in presence of annealed disorder (EW and KPZ) in terms of both growth equations and discrete models. In Chapter 3 we focus on the elastic interface dynamics in disordered media, i.e., in presence of quenched randomness. This Chapter contains original research based on cellular automata simulations. We carry out a novel study of the dynamics by focusing on the discrete activity patterns that the interface sites describe during therelaxation toward the steady state. We analyze the spatio-temporal correlations of such patterns as the temperature is varied. We observe that, for some range of low temperatures, the out-of-equilibrium relaxation can be understood in the context of creep dynamics.The second part of the Thesis focus on Sociophysics. This discipline attends to the social interactions among individuals -most often mapped onto networks to provide them a topological structure- and has recently attracted much interest in the physics community. Social interactions give rise to adaptive systems that exhibit complex features as self-organization and cooperation. Therefore, Statistical Mechanics provides the necessary tools to analyze the behavior of such groups of agentsin a first level of simplification. The topics that Sociophysics deals with are quite a number, and we particularly focus on processes of opinion formation. The Chapter 4 presents a basic classification of the different opinion formation models present in the literature. In Chapter 5 we provide some analytical and numerical own results to describe the effect that the social temperature- understood as a simplified description of the interplay between an agent, its surroundings, and a collective climate parameter- may exert on such opinion formation processes. The thermal effect can be implemented in different ways. In the first part of the Chapter we work on a simple opinion formation model that, according to some procedural rules, reproduces the Sznajd dynamics. We include the thermal effect by means of some probability that the agents adopt the opposite opinion that the one indicated by such rules. In the second part of the Chapterwe consider a system with three different interacting groups of individuals, where the thermal effect is implemented as certain probability of spontaneous changes of the agents opinion. We exploit the van Kampen's expansion approach to analyze the macroscopic behavior of the different supporter group densities as well as the fluctuations around such macroscopic behavior.The third and last part of the document concerns Complex Networks, which have recently prompted the scientific community to investigate the mechanisms that determine their topology and dynamical properties.The rapid development of networks like the Internet and the World-Wide-Web, which represent today the basic substrate for all sort of communications at planetary level, has given rise to a number of interdisciplinary studies with highly technological applications. We first provide an introduction to complex networks in Chapter 6, where we introduce some basic concepts as scale-free graphs, mixing patterns, clustering coefficient, and small-world effect. In Chapter 7 we deal with traffic processes on networks, and specifically we focus on optimization of the routing protocols that define the connecting paths among all the pair of nodes. Such optimization pursues to avoid the traffic jams that emerge for huge quantities of matter or information flowing inthe graph. We propose an optimization algorithm that, in order to avert jamming, minimizes the number of paths that go through the most visited node (maximal betweenness) while keeping the path length as short as possible, i.e., in the proximities of the length distribution of the initial shortest-path protocol.

Molecular beam epitaxy has recently been applied to the growth and self assembly of nanostructures on crystal substrates. This highlights the importance of understanding how microscopic rules of atomic motion and assembly lead to macroscopic surface shapes. In this thesis, we present results from two computational studies of these mechanisms. We identify a kinetic mechanism responsible for the emergence of low-angle facets in recent epitaxial regrowth experiments on patterned surfaces. Kinetic Monte Carlo simulations of vicinal surfaces show that the preferred slope of the facets matches the threshold slope for the transition between step flow and growth by island nucleation. At this crossover slope, the surface step density is minimized and the adatom density is maximized, respectively. A model is developed that predicts the temperature dependence of the crossover slope and hence the facet slope. We also examine the "step bunching" instability thought to be present in step flow growth on surfaces with a downhill diffusion bias. One mechanism thought to produce the necessary bias is the inverse Ehrlich Schwoebel (ES) barrier. Using continuum, stochastic, and hybrid models of one dimensional step flow, we show that an inverse ES barrier to adatom migration is an insufficient condition to destabilize a surface against step bunching.

Molecular beam epitaxy has recently been applied to the growth and self assembly of nanostructures on crystal substrates. This highlights the importance of understanding how microscopic rules of atomic motion and assembly lead to macroscopic surface shapes. In this thesis, we present results from two computational studies of these mechanisms. We identify a kinetic mechanism responsible for the emergence of low-angle facets in recent epitaxial regrowth experiments on patterned surfaces. Kinetic Monte Carlo simulations of vicinal surfaces show that the preferred slope of the facets matches the threshold slope for the transition between step flow and growth by island nucleation. At this crossover slope, the surface step density is minimized and the adatom density is maximized, respectively. A model is developed that predicts the temperature dependence of the crossover slope and hence the facet slope. We also examine the "step bunching" instability thought to be present in step flow growth on surfaces with a downhill diffusion bias. One mechanism thought to produce the necessary bias is the inverse Ehrlich Schwoebel (ES) barrier. Using continuum, stochastic, and hybrid models of one dimensional step flow, we show that an inverse ES barrier to adatom migration is an insufficient condition to destabilize a surface against step bunching.

Nesse trabalho, são investigadas as propriedades dinâmicas de modelos da mecânica estatística na criticalidade. Inicialmente, trabalhando com modelos de spin e utilizando os conceitos de persistência global e de dimensão anômala da magnetização inicial, mostramos que o modelo de Baxter-Wu não está na mesma classe de universalidade dos modelos de Potts com quatro estados e de Ising com interação de três spins em uma direção, todos bidimensionais. Na segunda parte da tese, estudamos o fenômeno de crescimento da superfície gerada pela deposição segundo as regras que definem os autômatos celulares probabilísticos propostos por Grassberger (modelos A e B). Esses dois autômatos não pertencem à classe de universalidade de Domany-Kinzel e apresentam novos expoentes críticos, cuja origem se deve à conservação de paridade. Determinamos o expoente de crescimento beta w, válido em tempos curtos, assim como os outros expoentes críticos associados ao crescimento de superfície (alfa e z). Nossas estimativas se comparam bem com os resultados obtidos a partir de razões de inteiros propostas por Jensen para os expoentes beta, ni paralelo e ni perpendicular. Finalmente, investigamos a transição de fase entre o estado helicoidal e o estado desordenado (random coil) da polialanina e do fragmento peptídico PTH(1-34), que corresponde aos resíduos 1 a 34 da região aminoterminal do hormônio das paratireóides. Nosso cálculo, que leva em conta as interações entre todos os átomos da molécula, está baseado em uma abordagem de tempos curtos. Os resultados dessa análise indicam que a transição helix-coil das polialaninas e do PTH(1-34) é de segunda ordem e apontam para uma classe de universalidade para a transição helix-coil em homopolímeros e proteínas (partindo de um estado helicoidal). / In this work we investigated dynamic properties of statistical mechanical models at criticality. At first, using the concepts of global persistence and anomalous dimension of initial magnetization, we showed that the Baxter-Wu model does not belong to the same universality class as 4-state Potts model and Ising with multispin interaction in one direction. In the sequence, we studied the roughening behavior generated by deposition governed by rules defined by probabilistic cellular automata proposed by Grassberger (A and B models). Those models are known do not belong to the Domany-Kinzel universality class. They are characterized by different exponents which are related to the parity conserving (PC). We estimated the growth exponent beta w, in short-time regimen, such as, other critical exponents associated to the surface growth (alpha and z). Our results are in good agreement with those expected for parity conserving universality class. At last we studied the phase transition between the completely helical state and the random coil of the polyalanine, such as, for the 34-residue human parathyroid fragment PTH(1-34). Our short-time simulations of the helix-coil transition are based on a detailed all-atom representation of proteins. The results indicate that helix-coil transition in polyalanine and PTH(1-34) is a second-order phase transition and suggest a universality class to the helix-coil transition in homopolymer and (helical) proteins.

Molecular beam epitaxy has recently been applied to the growth and self assembly of nanostructures on crystal substrates. This highlights the importance of understanding how microscopic rules of atomic motion and assembly lead to macroscopic surface shapes. In this thesis, we present results from two computational studies of these mechanisms. We identify a kinetic mechanism responsible for the emergence of low-angle facets in recent epitaxial regrowth experiments on patterned surfaces. Kinetic Monte Carlo simulations of vicinal surfaces show that the preferred slope of the facets matches the threshold slope for the transition between step flow and growth by island nucleation. At this crossover slope, the surface step density is minimized and the adatom density is maximized, respectively. A model is developed that predicts the temperature dependence of the crossover slope and hence the facet slope. We also examine the "step bunching" instability thought to be present in step flow growth on surfaces with a downhill diffusion bias. One mechanism thought to produce the necessary bias is the inverse Ehrlich Schwoebel (ES) barrier. Using continuum, stochastic, and hybrid models of one dimensional step flow, we show that an inverse ES barrier to adatom migration is an insufficient condition to destabilize a surface against step bunching. / Science, Faculty of / Physics and Astronomy, Department of / Graduate